Abstract

Many properties of the moduli space of abelian vortices on a compact Riemann surface Σ are known. For non-abelian vortices the moduli space is less well understood. Here we consider non-abelian vortices on the Riemann sphere , and we study their moduli spaces near the Bradlow limit. We give an explicit description of the moduli space as a Kähler quotient of a finite-dimensional linear space. The dimensions of some of these moduli spaces are derived. Strikingly, there exist non-abelian vortex configurations on , with non-trivial vortex number, for which the moduli space is a point. This is in stark contrast to the moduli space of abelian vortices. For a special class of non-abelian vortices the moduli space is a Grassmannian, and the metric near the Bradlow limit is a natural generalization of the Fubini–Study metric on complex projective space. We use this metric to investigate the statistical mechanics of non-abelian vortices. The partition function is found to be analogous to the one for abelian vortices.

Received 12 January 2013Accepted 06 March 2013Published online 04 April 2013

Acknowledgments:

This work was carried out as part of my Ph.D. research. I wish to thank my research supervisor, Nick Manton, for numerous helpful discussions and for comments on the manuscript of this paper. I acknowledge two discussions with Martin Speight, which gave me some motivation to continue my research on non-abelian vortices, but did not lead to any of the results presented here. I am indebted to Julian Holstein, Marco Golla, Peter Herbrich, and John Ottem for many discussions on aspects of Algebraic Topology and Algebraic Geometry, and to Martin Wolf and Moritz Högner for discussions on holomorphic vector bundles. I thank Daniele Dorigoni for a discussion on Yang–Mills–Higgs models with different numbers of colours and flavours. This work was financially supported by the EPSRC, the Cambridge European Trust, and St. John's College, Cambridge.